Electrical and Optical Properties of Materials Part 4: Maxwell's Equations

Electrical and optical properties of materials JJL Morton

Electrical and optical properties of materials John JL Morton Part 4: Maxwell’s Equations We have already used Maxwell’s equations for electromagnetism, and in many ways they are simply a reformulation (or even just a copy) of equations you have already come across and which were established before Maxwell. However, the assimilation of a set of existing knowledge in a particular way can be extremely valuable. Furthermore, Maxwell derived three of these equations independently, and in doing so made a critical correction which makes possible the electromagnetic wave. We shall see that Maxwell’s equations will unify electricity, magnetism and even light. In this regard, it is entirely appropriate that we know the set as ‘his’ equations.

4.1 A reminder on waves

E E time = 0 time = t z0 vt z0

z z

Figure 4.1: Travelling waves

A travelling wave has some arbitrary wave-form which remains fixed in shape and propagates through space at some velocity v. A ‘real-world’ wave has some amplitude representing some physical property (e.g. electric field E) which becomes a function of both distance and time. Suppose the wave propagates along the z axis (see Figure 4.1). At time t = 0 the amplitude of E has some form E = f(z). At time t the waveform has advanced a distance vt along z. The equation now representing the same wave has shifted: E = f(z − vt) (4.1) This general equation describes any non-decaying wave. Defining u = z − vt, we use the product rule to differentiate E with respect to z: ∂E ∂f ∂u ∂u = , and = 1 (4.2) ∂z ∂u ∂z ∂z Repeating this differentiation, with respect to z, ∂2E ∂2f ∂u ∂2f = = (4.3) ∂z2 ∂u2 ∂z ∂u2 1 4. Maxwell’s equations for electromagnetism

Then we do the same with respect to time: ∂E ∂f ∂u ∂u = , here = −v (4.4) ∂t ∂u ∂t ∂t Repeating this diﬀerentiation, with respect to z,

∂2E ∂2f ∂u ∂2f = −v = v2 (4.5) ∂t2 ∂u2 ∂t ∂u2 Combing Eq. 4.3 and 4.5 above gives us the wave equation:

∂2E 1 ∂2E = (4.6) ∂z2 v2 ∂t2 Waves with sine or cosine amplitude are solutions to the above equation (reassuringly!). For example, take this cosine wave with wavelength λ:

2π E = E cos (z − vt) (4.7) 0 λ

Deﬁning the wavenumber k = 2π/λ, and remembering that the velocity v = fλ where f is the frequency:

E = E0 cos (kz − 2πft) (4.8)

We often use the angular frequency ω = 2πf (it has units of radians per second). E = E0 cos (kz − ωt) (4.9) Plugging this into the wave equation Eq. 4.6 gives: 1 k2E = ω2E or v = ω/k (4.10) v2 Finally, remember that we can use exp(iφ) = cos(φ) + i sin(φ) as a combina- tion of cosine and sine solutions for a general wave. Thus, deﬁning a general wave as E = E0 exp[i(kx − ωt)] is quite common.

4.2 Deﬁning Maxwell’s equations

4.2.1 Maxwell 1 and 2 The first two Maxwell’s equations are really a formulation of Guass’s Theo- rem for electric and magnetic ‘charges’, and are best appreciated using the picture of vector fields. We are used to drawing vector fields to describe

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Figure 4.2: A vector ﬁeld

Figure 4.3: Left and centre: a positive or negative electric charge will act as a sink or source of electric field lines. The total electric field flux through the surface (red) is equal to the charge contained within (Maxwell 1). The same applies for magnetic fields, but as we always have magnetic dipoles, all surfaces will have net magnetic field flux of 0 (Maxwell 2).

how a field is oriented and varies through space (e.g. wind), as shown in Figure 4.2. Maxwell’s first equation states that the net flux of electric field E through any closed surface is equal to the net charge contained inside, di- vided by the permittivity . This is illustrated in Figure 4.3 for positive and negative charges, which act as sinks or sources for an electric field. However, if we try to make the same picture for a magnetic dipole, we see that the net flux through any surface is always zero (precisely because it is a dipole, not a monopole). No magnetic monopoles or magnetic ‘charges’ exist — or more accurately have yet been found. Hence, Maxwell’s second equation is as above, but for magnetic fields where the magnetic ‘charge’ contained is always 0.

These statements can be formalised mathematically by taking the sum of the derivative of each component of the ﬁeld along each direction x, y and z.

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Maxwell’s Equations

1 The ﬂux of E through any closed surface is equal to the net charge contained inside /

∂Ex + ∂Ey + ∂Ez = ρ/ ∂x ∂y ∂z 2 The ﬂux of B through any closed surface is zero

∂Bx + ∂By + ∂Bz = 0 ∂x ∂y ∂z

4.2.2 Maxwell 3 and 4 We live in a three dimensional world (well arguably more...), and so it is clear that any general theory of electromagnetism must handle electric and magnetic ﬁelds which point and propagate in any direction. Maxwell’s next two equations in their general form do this, however this requires a mathe- matical toolbox (namely the vector operators div, grad and curl) which lie beyond the scope of this course. However, we can probe the richness of these equations and their conclusions in a set of restricted dimensions. Let us state that the electric and magnetic ﬁelds are only allowed variation in time and along a direction z (as would be the case for some plane-wave propagation in one direction).

Maxwell’s Equations in 1D (ish)

∂Ex ∂By 3 ∂z = − ∂t

∂By ∂Ex 4 ∂z = −µ Jx + ∂t

We can derive the next two of Maxwell’s equations for this case of propagation in one dimension, with the help of Figure 4.4. We have defined the electric field E to point along x, and the magnetic field B to point along y (Note1). We can begin by using Faraday’s Law of induction around the loop in red (x − z). Faraday’s Law states that the voltage around this loop is

1This doesn’t necessarily take away from the generality of the result — we are working out the behaviour of the magnetic ﬁeld orthogonal to the electric ﬁeld, which in general could point anywhere in the x − y plane.

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y B B+δB x E δx δy E+δE

wave propagates z δz this way Figure 4.4: Deriving Maxwell’s equations 3 and 4 for propagation in 1D equal to the rate of change of ﬂux through it:

δB (Eδx − (E + δE)δx) = y δxδz (4.11) δt Rearranging gives: δE δB x = − y (4.12) δz δt Now we’ll use Amp`eresLaw around the blue loop (y −z): this states that the integral of magnetic ﬁeld around a loop is equal to the current running through the loop, times µ, the permeability.

(Bδy − (B + δB)δy) = µJxδyδz (4.13)

δB y = −µJ (4.14) δz x We have multiplied the current density J by the area of the loop to get a current. Now, there is an all-important correction to Ampère’s Law which Maxwell introduced into this equation. He postulated that there was an ad- ditional current density term = ∂Ex/∂t which we now call the displacement current [2]. ∂B ∂E y = −µ J + x (4.15) ∂z x ∂t There was no a priori reason why it should be there, but in adding it Maxwell was able to define the electromagnetic wave, and the velocity with which it may travel. Experiments conducted years later by Hertz were able to confirm Maxwell’s insight and the significance of the relations he had derived.

2You may also see Maxwell’s equations in terms of D = E and H = B/µ. The displacement current is then ∂D/∂t

5 4. Maxwell’s equations for electromagnetism

4.3 Electromagnetic waves in a vacuum

We’ll begin by taking a look at Maxwell’s Equations 3 and 4 in a vacuum, where the current J = 0. We can take 3 and diﬀerentiate both sides with respect to z: ∂2E ∂ ∂B x = − y (4.16) ∂z2 ∂z ∂t ∂2E ∂ ∂B x = − y (4.17) ∂z2 ∂t ∂z Substitute in 4 on the right hand side: ∂2E ∂ ∂E x = µ x (4.18) ∂z2 ∂t 0 0 ∂t ∂2E ∂2E x = µ x (4.19) ∂z2 0 0 ∂t2 Looking back at Eq. 4.6 we see that this is nothing else than the wave equation. Thus, electric and magnetic ﬁelds propagate as waves with velocity: 1 1 2 = µ00 or v = √ (4.20) v µ00

The values of µ0 and 0 were known at the time of Maxwell’s work from measurements by Amp`ere,Faraday and others: −7 −1 −12 2 −1 −1 µ0 = 4π × 10 Hm and 0 = 8.854 × 10 s H m (4.21) Thus, Maxwell was able to calculate the speed with which electromagnetic waves travel through a vacuum (or ‘free space’):

1 −1 c0 = √ = 299 792 458 ms (4.22) µ00

4.4 Electromagnetic waves in non-conducting, uncharged dielectrics

The derivation above assumed the simplest case of a vacuum. We can gener- alise this somewhat to a non-conducting (J remains 0) material with relative permittivity r and permeability µr. The same steps give: 1 c c = √ = √ 0 (4.23) µ0µr0r rµr

As r and µr are always greater than 1, this means light never travels faster than in a vacuum. The refractive index n of a material describes by what factor light slows down in materials: c √ n = 0 = µ (4.24) c r r

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4.5 Impedance

The impedance Z is defined as the ratio of the strengths of the electric field 3 Ex and magnetizing field Hy (= By/µ) [Note ]. Now that we know they are waves, we can call the electric field Ex = E0 exp(i(kz − ωt)), and the magnetic field By = B0 exp(i(kz − ωt)). From Maxwell 3 we have:

ikE0 exp(i(kz − ωt)) = iωB0 exp(i(kz − ωt)) (4.25) E ω 0 = = c (4.26) B0 k The impedance is thefore: E µE µ rµ Z = x = 0 = µc = √ = (4.27) Hy B0 µ The impedance of free space is: r µ0 Z0 = = 377 Ω (= 120π Ω is a common approx.) (4.28) 0 Similarly, in an isotropic, charge-less material, the characteristic impedance is: r µ0µr Z = µrµ0c = (4.29) 0r

4.6 Electromagnetic waves in conducting dielectrics

For a conducting dielectric we can no longer assume the current density is zero. We must therefore go through the derivation in Eqs 4.16 through 4.19 again, without neglecting J: ∂2E ∂ ∂B x = − y (4.30) ∂z2 ∂z ∂t ∂2E ∂ ∂B x = − y (4.31) ∂z2 ∂t ∂z Substitute in Maxwell 4 on the right hand side: ∂2E ∂ ∂E x = µ J + x (4.32) ∂z2 ∂t x ∂t

3Why do we use the magnetizing ﬁeld H and not the magnetic ﬁeld B? The units of H are amps per metre and those of E are volts per metre. The ratio of E and H therefore has units of volts per amp (or ohms) the units for an impedance

7 4. Maxwell’s equations for electromagnetism

In a conducting medium, the current density J is related to the applied electric ﬁeld E by the relation J = σE: ∂2E ∂ ∂E x = µ σE + x (4.33) ∂z2 ∂t x ∂t ∂2E ∂E ∂2E x = µσ x + µ x (4.34) ∂z2 ∂t ∂t2 This resulting equation is similar to the wave equation. Let’s therefore try a trial solution with the same form of wave: E = E0 exp[i(kz − ωt)]. Plugging it in, we see that it is indeed a solution, with: p k = µω2 + iωµσ (4.35) For σ = 0 we obtain the same results as derived above for non-conducting media. However, we see that for non-zero σ, the wavenumber k acquires some imaginary component. This imaginary component results in a damping of the wave with distance. For complex k,

i(kz−ωt) i(Re[k]z−ωt) −Im[k]z E = E0e = E0e e (4.36) A very reasonable assumption for a metal is that the conduction current (which goes as σ) is much greater than the displacement current (which goes as ω). Thus, for a good conductor we can say that σ >> ω and the above relation for δ simplifies to4: rσµω Re[k] = Im[k] = (4.37) 2 The decay constant of the damping with distance is thus 1/Im[k]. We call this decay constant the skin depth δ: r 2 δ = (4.38) σµω 8 −1 Some examples: Copper has σCu = 10 Sm , µr = 1, so for 1 MHz the skin depth is only δ = 65 µm. At 10 GHz it falls to 650 nm. Materials with small skin depths therefore become ineffective at transmitting electromagnetic waves at high frequencies and the wave only travels along the outer ‘skin’ of the conductor and the effective resistance increases. High voltage overhead transmission lines often use aluminium cable (with a steel reinforcing core) for this reason. The steel core has high-resistivity and can be neglected — due to the skin effect the current is carried primarily by the outer aluminium portion of the cable δAl @ 50 Hz= 11 mm. √ √ 4Reminder: i = (1 + i)/ 2

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4.7 Energy ﬂow and the Poynting vector

The energy stored (per unit volume) in electromagnetic ﬁelds is: 1 1 U = (D · E + B · H) = (E2 + µH2), (4.39) 2 2 x y in the case of our wave propagating in the z direction. Give a wave velocity c, the energy ﬂux (per unit area) is:

1 r rµ 1 E2 cU = E2 + H2 = x + ZH2 , (4.40) 2 µ x y 2 Z y where Z is the impedance derived above. We deﬁned the impedance to be the ratio of Ex to Hy, so

E2 flux = cU = E H = x = ZH2 (4.41) x y Z y We see the energy is stored equally in the electric and magnetic fields. The Poynting vector gives a measure of the direction and flux of energy flow in electromagnetic fields:

P = E × H (Pz = ExHy) (4.42)